A family of iterative methods that uses divided differences of first and second orders

The family of fourth-order Steffensen-type methods proposed by Zheng et al. (Appl. Math. Comput. 217, 9592-9597 (2011)) is extended to solve systems of nonlinear equations. This extension uses multidimensional divided differences of first and second orders. For a certain computational efficiency index, two optimal methods are identified in the family. Semilocal convergence is shown for one of these optimal methods under mild conditions. Moreover, a numerical example is given to illustrate the theoretical results.Peer ReviewedPostprint (author's final draft

CMMSE-2019 mean-based iterative methods for solving nonlinear chemistry problems

[EN] The third-order iterative method designed by Weerakoon and Fernando includes the arithmetic mean of two functional evaluations in its expression. Replacing this arithmetic mean with different means, other iterative methods have been proposed in the literature. The evolution of these methods in terms of order of convergence implies the inclusion of a weight function for each case, showing an optimal fourth-order convergence, in the sense of Kung-Traub's conjecture. The analysis of these new schemes is performed by means of complex dynamics. These methods are applied on the solution of the nonlinear Colebrook-White equation and the nonlinear system of the equilibrium conversion, both frequently used in Chemistry.This research was partially supported by PGC2018-095896-B-C22 (MCIU/AEI/FEDER/UE) and Generalitat Valenciana PROMETEO/2016/089.Chicharro, FI.; Cordero Barbero, A.; Martínez, TH.; Torregrosa Sánchez, JR. (2020). CMMSE-2019 mean-based iterative methods for solving nonlinear chemistry problems. Journal of Mathematical Chemistry. 58(3):555-572. https://doi.org/10.1007/s10910-019-01085-2S555572583O. Ababneh, New Newton’s method with third order convergence for solving nonlinear equations. World Acad. Sci. Eng. Technol. 61, 1071–1073 (2012)S. Amat, S. Busquier, Advances in iterative methods for nonlinear equations, chapter 5. SEMA SIMAI Springer Series. (Springer, Berlin, 2016), vol. 10, pp. 79–111R. Behl, Í. Sarría, R. González, Á.A. Magreñán, Highly efficient family of iterative methods for solving nonlinear models. J. Comput. Appl. Math. 346, 110–132 (2019)B. Campos, J. Canela, P. Vindel, Convergence regions for the Chebyshev-Halley family. Commun. Nonlinear Sci. Numer. Simul. 56, 508–525 (2018)F.I. Chicharro, A. Cordero, J.R. Torregrosa, Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 780513, 1–11 (2013)F.I. Chicharro, A. Cordero, J.R. Torregrosa, Dynamics of iterative families with memory based on weight functions procedure. J. Comput. Appl. Math. 354, 286–298 (2019)C.F. Colebrook, C.M. White, Experiments with fluid friction in roughened pipes. Proc. R. Soc. Lond. 161, 367–381 (1937)A. Constantinides, N. Mostoufi, Numerical Methods for Chemical Engineers with MATLAB Applications (Prentice-Hall, Englewood Cliffs, 1999)A. Cordero, J. Franceschi, J.R. Torregrosa, A.C. Zagati, A convex combination approach for mean-based variants of Newton’s method. Symmetry 11, 1062 (2019)A. Cordero, J.R. Torregrosa, Variants of Newton’s method using fifth order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Math. 21, 643–651 (1974)T. Lukić, N. Ralević, Geometric mean Newton’s method for simple and multiple roots. Appl. Math. Lett. 21, 30–36 (2008)A. Özban, Some new variants of Newton’s method. Appl. Math. Lett. 17, 677–682 (2004)M. Petković, B. Neta, L. Petković, J. Dz̆unić, Multipoint Methods for Solving Nonlinear Equations (Academic Press, Cambridge, 2013)E. Shashi, Transmission Pipeline Calculations and Simulations Manual, Fluid Flow in Pipes (Elsevier, London, 2015), pp. 149–234M.K. Singh, A.K. Singh, A new-mean type variant of Newton’s method for simple and multiple roots. Int. J. Math. Trends Technol. 49, 174–177 (2017)K. Verma, On the centroidal mean Newton’s method for simple and multiple roots of nonlinear equations. Int. J. Comput. Sci. Math. 7, 126–143 (2016)S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with third-order convergence. Appl. Math. Lett. 13, 87–93 (2000)Z. Xiaojian, A class of Newton’s methods with third-order convergence. Appl. Math. Lett. 20, 1026–1030 (2007